Problem: What is the slope of the line tangent to $f(x) = -2x^{2}-2x+6$ at $x = 2$ ?
Explanation: The slope of the tangent line is $ \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$ $ = \lim_{h \to 0} \frac{(-2(x+h)^{2}-2(x+h)+6) - (-2x^{2}-2x+6)}{h}$ $ = \lim_{h \to 0} \frac{(-2(x^{2}+2x h+h^{2})-2(x+h)+6) - (-2x^{2}-2x+6)}{h}$ $ = \lim_{h \to 0} \frac{-2x^{2}-4(x h)-2h^{2}-2x-2h+6+2x^{2}+2x-6}{h}$ $ = \lim_{h \to 0} \frac{-4(x h)-2h^{2}-2h}{h}$ $ = \lim_{h \to 0} -4x-2h-2$ $ = -4x-2$ $ = (-4)(2)-2$ $ = -10$